THÈSE Estimation, validation et identification des modèles ARMA ...

Chapitre 5. Model selection of weak VARMA models 157
5.3 General multivariate linear regression model
Let Zt = (Z1t,...,Zdt) ′ be a d-dimensional random vector of response variables,
Xt = (X1t,...,Xkt) ′ be a k-dimensional input variables and B = (β1,...,βd) be a
k × d matrix. We consider a multivariate linear model of the form Zit = X ′ tβi + ǫit,
i = 1,...,d, orZ ′ t = X′ tB+ǫ′ t ,t = 1,...,n, where theǫt = (ǫ1t,...,ǫdt) ′ are uncorrelated
and identically distributed random vectors with variance Σ = Eǫtǫ ′ t . The i-th column
of B (i.e. βi) is the vector of regression coefficients for the i-th response variable.
Now, given the n observations Z1,...,Zn and X1,...,Xn, we define the n × d data
matrix Z = (Z1,...,Zn) ′ , the n × k matrix X = (X1,...,Xn) ′ and the n × d matrix
ε = (ǫ1,...,ǫn) ′ . Then, we have the multivariate linear model Z = XB +ε. Now, it is
well known that the QMLE of B is the same as the LSE and, hence, is given by
ˆB = (X ′ X) −1 X ′ Z, that is, ˆ βi = (X ′ X) −1 X ′ Zi, i = 1,...,d,
where Zi = (Zi 1,...,Zi n) ′ is the i-th column of Z. We also have
ˆε := Z−X ˆ B = MXZ = ε−X(X ′ X) −1 X ′ ε = MXε,
where MX = In −X(X ′ X) −1 X ′ is a projection matrix. The usual unbiased estimator
of the error covariance matrix Σ is
Σ ∗ =
=
1
n−k ˆε′ ˆε = 1
n−k (Z−Xˆ B) ′ (Z−X ˆ B)
n 1
(Zt −
n−k
ˆ B ′ Xt)(Zt − ˆ B ′ Xt) ′
t=1
or Σ∗ = (n − k) −1n t=1ˆǫtˆǫ ′ t , where the ˆǫt = Zt − ˆ B ′ Xt are the residual vectors. Note
that the gaussian quasi-likelihood is given by
1
Ln(B,Σ;Z) =
(2π) d/2√
exp −
detΣe
1
n
(Zt −B
2
′ Xt) ′ Σ −1 (Zt −B ′
Xt) ,
whose maximization shows that the QML estimators of B is equal to ˆ B and that of Σ is
ˆΣ := n −1 n
t=1 ˆǫtˆǫ ′ t = (n−k)n−1 Σ ∗ . Because Σ ∗ is an unbiased estimator of the matrix
Σ, by definition we have E{Σ∗ } = Σ, we then deduce that
n
n−k E
ˆΣ
1
=
n−k Eˆε′ ˆε = 1
n−k Eε′ MXε = Σ. (5.3)
5.4 Kullback-Leibler discrepancy
Assume that, with respect to a σ-finite measure µ, the true density of the observationsX
= (X1,...,Xn) isf0, and that some candidate modelmgives a densityfm(·,θm)
t=1